The development of hybrid (HEV) and electric vehicles (EV) is a complex task that involves many new technologies that have not been used previously in the automotive industry. Mastering this task requires early testing of individual components in order to timely detect problems and to ensure seamless integration. However, components interact with each other such that isolated testing is impossible without a proper test bed that can emulate these interactions. For each component, an individual environment that matches its future operating conditions as closely as possible is required. A powertrain that contains one or more electric motors does not only require a mechanical but also an electrical testing environment. An important part of the electrical environment is the traction battery. Testing with a real battery requires time consuming pre-conditioning of the battery in order to achieve defined operating conditions for the powertrain. In addition, battery aging prevents deterministic repetition of test runs. Even more, it is often desirable to test the powertrain as early as possible when a suitable battery may not even be available yet. These challenges can be met by using a battery emulator (BE) that acts as a substitute for the traction battery by emulating its electrical characteristics. These are usually simulated using a more or less complex battery model. A programmable DC power supply replicates the simulated battery voltage at its output terminals and supplies the powertrain with the necessary power. The measured load current is fed back to the simulation model in order to update the state of the virtual battery. Because of the power levels of several tens or even hundreds of kilowatts, a switch-mode DC-DC converter has to be used instead of a linear power supply. The time constants of the electrochemical processes inside the battery are usually slow compared to that of the power electronics and the electric powertrain. Neverthe-less, because of capacitances due to double layer effects and ohmic resistance and inductance of interconnections between the cells, the terminal voltage can change very fast due to load current transients. E.g. a step increase in the load current will cause an immediate drop of the terminal voltage. It is not sufficient to just replicate the open circuit voltage from the battery model, the internal impedance of the battery must be emulated. As a consequence, it is necessary to design a controller that achieves fast output voltage reference tracking and effective load current rejection such that the output impedance of the battery emulator's output converter is suppressed and the impedance from the battery model can be imposed. Increasing demand for emulation of ultracapacitor batteries further increases the required bandwidth, because they exhibit faster dynamics than electrochemical batteries.
During operation, the traction inverter controls the speed or torque produced by the electric traction motor. Changes in the battery terminal voltage are compensated for by the controller such that the inverter's power output does not change. In the literature, such a load is usually called a constant power load (CPL). When a CPL is supplied by a power electronic converter instead of a battery, the system can become unstable due to the load's negative impedance characteristic. This is particularly problematic with compact, high performance automotive inverters. Compared to inverters for industrial applications, traction inverters usually have a small DC-link capacitor, which reduces the stability margin. As a further consequence of the resulting small filter capacitance, load transients and current ripple are propagated back to the DC-supply.
Power supply emulation has many fields of application. The emulation of batteries is also helpful for testing of consumer electronic devices, such as described in [P. H. Chou, C. Park, J. Park, K. Pham, and J. Liu, “B#: a battery emulator and power profiling instrument,” in ISLPED '03: Proceedings of the 2003 international symposium on Low power electronics and design. New York, N.Y., USA: ACM, 2003, pp. 288-293]. Testing of fuel cell power converters can be problematic due to the limited availability of fuel cell prototypes and the risk of expensive damage. Therefore, the emulation of fuel cells can be of advantage [A. Gebregergis and P. Pillay, “The development of solid oxide fuel cell (sofc) emulator,” in Power Electronics Specialists Conference, 2007. PESC 2007. IEEE, 17-21 2007, pp. 1232-1238]. In both documents, a linear power amplifier is used in order to interface the model of the power source to the system under test. Although such amplifiers can provide high bandwidths, they are limited to small power levels due to their low efficiency. Another important field of application is the testing of grid inverters for photovoltaic systems. In [M. C. Di Piazza and G. Vitale, “Photovoltaic field emulation including dynamic and partial shadow conditions,” Applied Energy, vol. 87, no. 3, pp. 814-823, 2010], a photovoltaic panel emulator based on a switch mode DC-DC converter is described. A description of an emulator for automotive starter batteries can be found in [T. Baumhöfer, W. Waag, and D. Sauer, “Specialized battery emulator for automotive electrical systems,” in Vehicle Power and Propulsion Conference (VPPC), 2010 IEEE, September 2010, pp. 1-4].
DC-DC converters connected to CPLs are studied in several publications. The concept of negative impedance instability was introduced by [V. Grigore, J. Hatonen, J. Kyyra, and T. Suntio, “Dynamics of a buck converter with a constant power load,” in Power Electronics Specialists Conference, 1998. PESC 98 Record. 29th Annual IEEE, vol. 1, 17-22 1998, pp. 72-78 vol. 1] and [B. Choi, B. Cho, and S. S. Hong, “Dynamics and control of dc-to-dc converters driving other converters downstream,” Circuits and Systems 1: Fundamental Theory and Applications, IEEE Transactions on, vol. 46, no. 10, pp. 1240-1248, October 1999]. The proposed stabilizing control approaches range from feedback linearization in [J. Ciezki and R. Ashton, “The application of feedback linearization techniques to the stabilization of dc-to-dc converters with constant power loads,” in Circuits and Systems, 1998. ISCAS '98. Proceedings of the 1998 IEEE International Symposium on, vol. 3, May-3, Jun. 1998, pp. 526-529 vol. 3] and [A. Emadi and M. Ehsani, “Negative impedance stabilizing controls for pwm dc-dc converters using feedback linearization techniques,” in Energy Conversion Engineering Conference and Exhibit, 2000. (IECEC) 35th Intersociety, vol. 1, 2000, pp. 613-620 vol. 1] over sliding mode control in [A. Emadi, A. Khaligh, C. Rivetta, and G. Williamson, “Constant power loads and negative impedance instability in automotive systems: definition, modeling, stability, and control of power electronic converters and motor drives,” Vehicular Technology, IEEE Transactions on, vol. 55, no. 4, pp. 1112-1125, July 2006] to passivity based PID design in [A. Kwasinski and P. Krein, “Passivity-based control of buck converters with constant-power loads,” in Power Electronics Specialists Conference, 2007. PESC 2007. IEEE, 2007, pp. 259-265] and active damping in [A. Rahimi and A. Emadi, “Active damping in dc/dc power electronic converters: A novel method to overcome the problems of constant power loads,” Industrial Electronics, IEEE Transactions on, vol. 56, no. 5, pp. 1428-1439, May 2009]. Model predictive control for stabilization of power systems containing CPLs is proposed in [M. Zima and G. Andersson, “Model predictive control employing trajectory sensitivities for power systems applications,” in Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on, 2005, pp. 4452-4456]. All of these have in common that a supply converter has to provide a constant voltage for one or more load converters that act as constant power loads. The proposed control design approaches lead to stable closed loops but the reference step responses (if considered at all) are slow and exhibit underdamped oscillations. In contrast, the application that is presented here requires fast voltage reference tracking.
Other power electronic converters that require fast reference tracking are uninterruptible power supplies (UPS) that generate an AC voltage. The periodic nature of the AC voltage can be exploited in order to improve reference tracking and disturbance rejection [K. Zhang, L. Peng, Y. Kang, and J. Xiong, “State-feedback-with-integral control plus repetitive control for UPS inverters,” in Twentieth Annual IEEE Applied Power Electronics Conference and Exposition, 2005. APEC 2005., no. 2. IEEE, 2005, pp. 553-559]. For a BE, the output voltage is not periodic and depends on the load current via the battery model so that these approaches cannot be applied here.
Due to the increasing computational power of digital controller platforms and improved algorithms, model predictive control (MPC) is not restricted to systems with slow dynamics any more. It can now also be applied to systems that require high sampling rates such as power electronic converters. The control of DC-DC converters with MPC is proposed by [T. Geyer, G. Papafotiou, and M. Morari, “On the optimal control of switch-mode dc-dc converters,” Hybrid Systems: Computation and Control, pp. 77-85, 2004] and experimental results are shown in [T. Geyer, G. Papafotiou, R. Frasca, and M. Morari, “Constrained optimal control of the step-down dc-dc converter,” Power Electronics, IEEE Transactions on, vol. 23, no. 5, pp. 2454-2464, September 2008], where the so-called explicit MPC (eMPC) [A. Bemporad, F. Borrelli, and M. Morari, “Model predictive control based on linear programming the explicit solution,” Automatic Control, IEEE Transactions on, vol. 47, no. 12, pp. 1974-1985, December 2002] was the key for computational feasibility. In [A. Wills, D. Bates, A. Fleming, B. Ninness, and R. Moheimani, “Application of mpc to an active structure using sampling rates up to 25 khz,” in Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC '05. 44th IEEE Conference on, 2005, pp. 3176-3181], experimental results for active vibration control using MPC with constraints at sampling rates from 5 kHz to 25 kHz are presented. Experimental results for nonlinear MPC of an isolated full-bridge converter with a sampling time of 150 μs are shown in [Y. Xie, R. Ghaemi, J. Sun, and J. Freudenberg, “Implicit model predictive control of a full bridge dc-dc converter,” Power Electronics, IEEE Transactions on, vol. 24, no. 12, pp. 2704-2713, 2009]. Nonlinear MPC of a boost converter is described in [J. Bonilla, R. De Keyser, M. Diehl, and J. ESPINOZA, “Fast NMPC of a DC-DC converter: an exact Newton real-time iteration approach,” in Proc. of the 7th IFAC Symposium on Nonlinear Control Systems (NOL-COS 2007), 2007], but no experimental results are given. The simulated online linear MPC of a three-phase grid inverter that is described in [S. Richter, S. Mariethoz, and M. Morari, “High-speed online mpc based on a fast gradient method applied to power converter control,” in American Control Conference (ACC), 2010, 302010-Jul. 2, 2010, pp. 4737-4743] is shown to be executable in 10 μs . . . 50 μs on a standard DSP, but no experimental results are presented. The algorithm that is used in said last document is based on the fast gradient method proposed posed in [S. Richter, C. Jones, and M. Morari, “Real-time input-constrained mpc using fast gradient methods,” in Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009. Proceedings of the 48th IEEE Conference on, 2009, pp. 7387-7393].
Fast QP solvers tailored for MPC are proposed in [R. Milman and E. Davison, “A fast mpc algorithm using nonfeasible active set methods,” Journal of Optimization Theory and Applications, vol. 139, pp. 591-616, 2008, 10.1007/s10957-008-9413-3], [H. J. Ferreau, H. G. Bock, and M. Diehl, “An online active set strategy to overcome the limitations of explicit mpc,” Int. J. Robust Nonlinear Control, vol. 18, no. 8, pp. 816-830, 2008] and [Y. Wang and S. Boyd, “Fast model predictive control using online optimization,” Control Systems Technology, IEEE Transactions on, vol. 18, no. 2, pp. 267-278, 2010].
Usually, controllers for DC-DC converters are designed for a nominal load, which in many cases is a resistor. Utilizing model based controller design it is possible to design a controller for a converter with an arbitrary load, as long as a suitable model is available. As a consequence, there is no need to rely on a nominal load resistor for control design. In this paper, we propose a converter model suitable for MPC design including a CPL with an additional input filter capacitance. The model is based on a linearized negative impedance approximation of the CPL which depends on the output voltage and the power demand of the load. Two different approaches for linear MPC design are proposed that can account for changes in the operating point. The first approach is a simple robust MPC design with two internal models that represent extremal values of the approximated load impedance. The second approach is a scheduling controller design that uses a set of different controller parameters for a number of operating points across the expected operating range. Based on an estimate of the load power demand, the closest parameter set is chosen for the calculation of the next control move at every sampling step. An observer is used in order to achieve offset free tracking despite unmeasured disturbances, filtering of measured disturbances and estimation of the load power demand.
For the implementation of constrained MPC, a heuristic active set method is proposed to quickly find a good active set within the limited time available for computation. This method and the robust MPC approach are also presented in [O. König, S. Jakubek, and G. Prochart, “Model predictive control of a battery emulator for testing of hybrid and electric power-trains,” 2011, accepted for presentation at: 2011 IEEE Vehicle Power and Propulsion Conference (VPPC)], however without constraints on the inductor current.
An approach to online model predictive control (MPC) of a high power step-down dc-dc converter is proposed. The converter is part of a battery emulator as a replacement for traction batteries on test beds for hybrid or fully electric automotive powertrains. This application requires fast tracking of a reference voltage from a simulated battery model, while being insensitive to fast load transients. The combination of the converter's weakly damped output filter with a tightly regulated load inverter that acts as a constant power load results in an unstable system. The exact specifications of the load are not known at the stage of control design and the power demand is fluctuating. Input constraint handling and inductor current limiting are required for optimal performance and for hardware protection. The proposed MPC based on an active set method achieves fast reference tracking despite the constant power load while respecting input and state constraints. The control algorithm can be executed at the required sampling rate on readily available digital controller hardware. Experimental results for a 60 kW battery emulator feeding an inverter demonstrate the performance of the proposed control approach.